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"r012:concept:maximum-degree", "definition": "The largest vertex degree occurring in a graph.", "prerequisite_concept_ids": [ "r012:concept:vertex-degree" ], "source_label": "maximum degree", "status": "source_frozen", "unit_ids": [ "r012:unit:s_graphs_exercises" ] }, { "concept_id": "r012:concept:maximum-flow", "definition": "A feasible network flow having greatest possible value.", "prerequisite_concept_ids": [ "r012:concept:network-flow", "r012:concept:flow-value" ], "source_label": "maximum flow", "status": "source_frozen", "unit_ids": [ "r012:unit:s_networkflow_flows-cuts" ] }, { "concept_id": "r012:concept:merge-sort", "definition": "A recursive sorting algorithm that sorts sublists and merges the resulting ordered lists.", "prerequisite_concept_ids": [ "r012:concept:sorting", "r012:concept:recursion" ], "source_label": "Merge Sort", "status": "source_frozen", "unit_ids": [ "r012:unit:s_induction_recursion_sorting", "r012:unit:s_posets_sorting" ] }, { "concept_id": "r012:concept:minimal-element", "definition": "An element having no strictly smaller element in the poset.", "prerequisite_concept_ids": [ "r012:concept:partial-order" ], "source_label": "minimal element", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_dilworth", "r012:unit:fig_height5" ] }, { "concept_id": "r012:concept:minimal-interval-representation", "definition": "The canonical integer-endpoint representation built from the ordered distinct down sets and up sets of an interval order.", "prerequisite_concept_ids": [ "r012:concept:interval-representation", "r012:concept:down-set", "r012:concept:up-set", "r012:concept:integer-endpoint" ], "source_label": "minimal interval representation", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_intervalorder_findrep", "r012:unit:thm_intord-minrep", "r012:unit:fig_10ptintorder" ] }, { "concept_id": "r012:concept:minimum-cut", "definition": "A source-sink cut having least possible capacity.", "prerequisite_concept_ids": [ "r012:concept:network-cut", "r012:concept:cut-capacity" ], "source_label": "minimum cut", "status": "source_frozen", "unit_ids": [ "r012:unit:s_networkflow_flows-cuts" ] }, { "concept_id": "r012:concept:minimum-spanning-tree", "definition": "A spanning tree whose total edge weight is minimum among all spanning trees of a weighted graph.", "prerequisite_concept_ids": [ "r012:concept:spanning-tree", "r012:concept:weighted-graph", "r012:concept:optimization" ], "source_label": "minimum-weight spanning tree", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_opt", "r012:unit:fig_spantree-1", "r012:unit:s_graphalgorithms_min-wt-span" ] }, { "concept_id": "r012:concept:minimum-spanning-tree-algorithm-history", "definition": "The publication and rediscovery lineage connecting Jarnik, Kruskal, Prim, and Dijkstra.", "prerequisite_concept_ids": [ "r012:concept:kruskal-algorithm", "r012:concept:prim-algorithm" ], "source_label": "history of minimum spanning-tree algorithms", "status": "source_frozen", "unit_ids": [ "r012:unit:s_graphalgorithms_historical-notes" ] }, { "concept_id": "r012:concept:modular-arithmetic", "definition": "Arithmetic in which integers are identified when they have the same remainder modulo a fixed positive integer.", "prerequisite_concept_ids": [], "source_label": "modular arithmetic", "status": "source_frozen", "unit_ids": [ "r012:unit:s_polya_apps" ] }, { "concept_id": "r012:concept:monotonicity-law", "definition": "A law describing how an operation preserves or reverses an order relation.", "prerequisite_concept_ids": [ "r012:concept:total-order" ], "source_label": "monotonicity law", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_prop-integers" ] }, { "concept_id": "r012:concept:multigraph", "definition": "A graph-like structure that permits loops and multiple edges.", "prerequisite_concept_ids": [ "r012:concept:graph", "r012:concept:multiple-edge", "r012:concept:graph-loop" ], "source_label": "multigraph", "status": "source_frozen", "unit_ids": [ "r012:unit:s_graphs_multigraphs" ] }, { "concept_id": "r012:concept:multinomial-coefficient", "definition": "The number of ways to divide a labeled n-element set into ordered groups of prescribed nonnegative sizes summing to n.", "prerequisite_concept_ids": [ "r012:concept:binomial-coefficient", "r012:concept:factorial" ], "source_label": "multinomial coefficient", "status": "source_frozen", "unit_ids": [ "r012:unit:s_strings_multinom", "r012:unit:exa_strings_multinom" ] }, { "concept_id": "r012:concept:multinomial-theorem", "definition": "The expansion of a nonnegative integer power of a sum of finitely many terms using multinomial coefficients.", "prerequisite_concept_ids": [ "r012:concept:multinomial-coefficient", "r012:concept:binomial-theorem" ], "source_label": "Multinomial Theorem", "status": "source_frozen", "unit_ids": [ "r012:unit:thm-strings-multinomial", "r012:unit:s_strings_lattice-paths-binomial-thm_activity" ] }, { "concept_id": "r012:concept:multiple-edge", "definition": "One of two or more distinct edges having the same pair of endpoints.", "prerequisite_concept_ids": [ "r012:concept:graph" ], "source_label": "multiple edge", "status": "source_frozen", "unit_ids": [ "r012:unit:s_graphs_multigraphs" ] }, { "concept_id": "r012:concept:multiplication-laws", "definition": "Left and right distributivity, associativity, identity, zero, and commutativity derived by induction.", "prerequisite_concept_ids": [ "r012:concept:recursive-multiplication", "r012:concept:addition-laws" ], "source_label": "laws of multiplication", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_mult" ] }, { "concept_id": "r012:concept:multiplication-monotonicity", "definition": "Multiplication by a nonnegative factor preserves weak order and by a positive factor preserves strict order.", "prerequisite_concept_ids": [ "r012:concept:natural-number-order", "r012:concept:integer-multiplication" ], "source_label": "monotonicity of multiplication", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_total-order-N" ] }, { "concept_id": "r012:concept:multiplicative-cancellation", "definition": "Equal products with the same nonzero factor have equal remaining factors.", "prerequisite_concept_ids": [ "r012:concept:multiplication-monotonicity" ], "source_label": "multiplicative cancellation", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_total-order-N" ] }, { "concept_id": "r012:concept:multiplicative-inverse", "definition": "For a nonzero fraction (a,b), the reciprocal (b,a), whose product with it is one.", "prerequisite_concept_ids": [ "r012:concept:rational-arithmetic" ], "source_label": "multiplicative inverse", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_rationals" ] }, { "concept_id": "r012:concept:multiset-generating-function", "definition": "The coefficient of x^k in (1-x)^(-n) counts size-k multisets from n types, equivalently distributions of k indistinguishable objects to n recipients.", "prerequisite_concept_ids": [ "r012:concept:generating-function-distribution-model", "r012:concept:binomial-coefficient" ], "source_label": "generating function for multisets", "status": "source_frozen", "unit_ids": [ "r012:unit:s_genfunction_distributions", "r012:unit:exa_genfunction_multisets", "r012:unit:auto:d7c45bbaac6cda937a07c30e" ] }, { "concept_id": "r012:concept:naive-recursion-cost", "definition": "Growth in execution time caused by repeatedly recomputing overlapping recursive subproblems.", "prerequisite_concept_ids": [ "r012:concept:recursive-evaluation", "r012:concept:computational-complexity" ], "source_label": "computational cost of naive recursive evaluation", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_discussion", "r012:unit:recurrence_discussion_takeaways" ] }, { "concept_id": "r012:concept:natural-number-order", "definition": "The relation m≤n defined by the existence of a nonnegative p with m+p=n.", "prerequisite_concept_ids": [ "r012:concept:nonnegative-integer", "r012:concept:integer-addition" ], "source_label": "order on nonnegative integers", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_total-order-N" ] }, { "concept_id": "r012:concept:natural-order-totality", "definition": "Reflexivity, antisymmetry, transitivity, and inductive comparability make the order total.", "prerequisite_concept_ids": [ "r012:concept:natural-number-order", "r012:concept:mathematical-induction" ], "source_label": "totality of natural-number order", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_total-order-N" ] }, { "concept_id": "r012:concept:necklace", "definition": "A circular arrangement of beads considered up to the symmetries specified by the problem.", "prerequisite_concept_ids": [ "r012:concept:enumeration" ], "source_label": "necklace", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_enum", "r012:unit:fig_intro_necklace" ] }, { "concept_id": "r012:concept:negated-relation", "definition": "Notation asserting that an ordered pair does not belong to a relation.", "prerequisite_concept_ids": [ "r012:concept:binary-relation", "r012:concept:logical-negation" ], "source_label": "negated relation membership", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_equivalence" ] }, { "concept_id": "r012:concept:negative-half-binomial-coefficient", "definition": "The closed form binom(-1/2,k)=(-1)^k binom(2k,k)/2^(2k).", "prerequisite_concept_ids": [ "r012:concept:generalized-binomial-coefficient", "r012:concept:falling-product-recurrence" ], "source_label": "negative one-half generalized binomial coefficient", "status": "source_frozen", "unit_ids": [ "r012:unit:l_newbinom", "r012:unit:auto:0bd3b62b200da00131d43e75" ] }, { "concept_id": "r012:concept:network-cut", "definition": "A partition V=L union U with the source in L and the sink in U.", "prerequisite_concept_ids": [ "r012:concept:flow-network" ], "source_label": "cut", "status": "source_frozen", "unit_ids": [ "r012:unit:s_networkflow_flows-cuts", "r012:unit:s_networkflow_exercises" ] }, { "concept_id": "r012:concept:network-flow", "definition": "Assignment of quantities to directed network edges subject to capacity and conservation constraints.", "prerequisite_concept_ids": [ "r012:concept:digraph", "r012:concept:optimization" ], "source_label": "network flow", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_intro", "r012:unit:ch_networkflow", "r012:unit:s_networkflow_intro", "r012:unit:s_networkflow_flows-cuts", "r012:unit:s_networkflow_augmenting", "r012:unit:s_networkflow_labeling-algorithm", "r012:unit:s_networkflow_example", "r012:unit:s_networkflow_integer-programming", "r012:unit:s_networkflow_exercises", "r012:unit:ch_flowapplications", "r012:unit:s_flowapplications_intro" ] }, { "concept_id": "r012:concept:network-flow-linear-program", "definition": "The formulation of maximum network flow as a linear optimization problem over edge-flow variables.", "prerequisite_concept_ids": [ "r012:concept:network-flow", "r012:concept:linear-programming" ], "source_label": "network flow as linear programming", "status": "source_frozen", "unit_ids": [ "r012:unit:s_networkflow_integer-programming" ] }, { "concept_id": "r012:concept:network-flow-optimality-certificate", "definition": "A cut whose capacity equals a feasible flow value, certifying that both are optimal.", "prerequisite_concept_ids": [ "r012:concept:flow-cut-capacity-bound" ], "source_label": "network-flow optimality certificate", "status": "source_frozen", "unit_ids": [ "r012:unit:s_networkflow_flows-cuts", "r012:unit:s_networkflow_labeling-algorithm", "r012:unit:s_networkflow_example" ] }, { "concept_id": "r012:concept:network-source-and-sink", "definition": "The distinguished initial and terminal vertices of a flow network; source edges point outward and sink edges point inward.", "prerequisite_concept_ids": [ "r012:concept:flow-network" ], "source_label": "source and sink", "status": "source_frozen", "unit_ids": [ "r012:unit:s_networkflow_intro" ] }, { "concept_id": "r012:concept:noncommutative-operation", "definition": "A binary operation for which changing the order of the operands can change the result.", "prerequisite_concept_ids": [], "source_label": "noncommutative operation", "status": "source_frozen", "unit_ids": [ "r012:unit:s_polya_perm-groups" ] }, { "concept_id": "r012:concept:nonconstant-coefficient-recurrence", "definition": "A linear recurrence equation in which one or more coefficients may depend on the sequence index.", "prerequisite_concept_ids": [ "r012:concept:linear-recurrence-equation" ], "source_label": "linear recurrence equation with nonconstant coefficients", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_linear", "r012:unit:rq-recurrence_linear-matching" ] }, { "concept_id": "r012:concept:nonhomogeneous-forcing-generating-function", "definition": "The generating function for the nonzero right-hand-side sequence, adjusted for the recurrence's starting index and combined with shifted solution-series terms.", "prerequisite_concept_ids": [ "r012:concept:generating-function-recurrence-method", "r012:concept:nonhomogeneous-linear-recurrence" ], "source_label": "generating function of the forcing sequence in a nonhomogeneous recurrence", "status": "source_frozen", "unit_ids": [ "r012:unit:ex_recurrence_gf-nonhomog" ] }, { "concept_id": "r012:concept:nonhomogeneous-linear-recurrence", "definition": "A linear recurrence equation whose forcing function is not identically zero.", "prerequisite_concept_ids": [ "r012:concept:linear-recurrence-equation" ], "source_label": "nonhomogeneous linear recurrence equation", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_linear", "r012:unit:rq-recurrence_linear-matching", "r012:unit:s_recurrence_solving_nonhomogeneous", "r012:unit:auto:4a09aa7241b0bf6a931d0955", "r012:unit:auto:2b9501b6517e77e868e24689", "r012:unit:ex_recurrence_lines-solved", "r012:unit:auto:025bf0a72a4db690049c74bf", "r012:unit:rq-recurrence_solving-async-nonlinear" ] }, { "concept_id": "r012:concept:nonincreasing-sequence", "definition": "A sequence whose successive terms never increase.", "prerequisite_concept_ids": [], "source_label": "non-increasing sequence", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_enum", "r012:unit:s_kitchensink_zero-one-matrices" ] }, { "concept_id": "r012:concept:nonisomorphic-graph", "definition": "Graphs for which no adjacency-preserving bijection exists between their vertex sets.", "prerequisite_concept_ids": [ "r012:concept:graph-isomorphism" ], "source_label": "nonisomorphic graphs", "status": "source_frozen", "unit_ids": [ "r012:unit:s_polya_apps", "r012:unit:s_polya_exercises" ] }, { "concept_id": "r012:concept:nonnegative-integer", "definition": "An integer greater than or equal to zero.", "prerequisite_concept_ids": [], "source_label": "non-negative integer", "status": "source_frozen", "unit_ids": [ "r012:unit:thm_ErdosSzekeres", "r012:unit:s_strings_comb-proof", "r012:unit:thm_binomial", "r012:unit:exa_inclusion-exclusion_int-solns", "r012:unit:auto:404b9eaf4b1ef3eb26f316ad", "r012:unit:auto:8f3500f97b0ab7c55c94505a", "r012:unit:s_background_int-union", "r012:unit:s_background_formal-num-sys", "r012:unit:s_background_add", "r012:unit:s_background_mult", "r012:unit:s_background_exp" ] }, { "concept_id": "r012:concept:notation-list", "definition": "An automatically generated reference list of mathematical notation used in the book.", "prerequisite_concept_ids": [ "r012:concept:back-matter" ], "source_label": "list of notation", "status": "source_frozen", "unit_ids": [ "r012:unit:auto:890b13ffad581b51ada6044f", "r012:unit:backmatter" ] }, { "concept_id": "r012:concept:np-class", "definition": "The class of decision problems whose yes instances have certificates verifiable in polynomial time.", "prerequisite_concept_ids": [ "r012:concept:polynomial-time" ], "source_label": "NP", "status": "source_frozen", "unit_ids": [ "r012:unit:s_basics_exact" ] }, { "concept_id": "r012:concept:number-system-construction-sequence", "definition": "The staged construction of natural, integer, rational, real, and complex number systems.", "prerequisite_concept_ids": [ "r012:concept:set-theoretic-foundations" ], "source_label": "number-system construction sequence", "status": "source_frozen", "unit_ids": [ "r012:unit:app_background" ] }, { "concept_id": "r012:concept:number-theory", "definition": "The study of integers and their arithmetic properties.", "prerequisite_concept_ids": [ "r012:concept:positive-integer" ], "source_label": "number theory", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_number", "r012:unit:s_inclusion-exclusion_euler-phi", "r012:unit:s_kitchensink_arithmetic-combinatorics" ] }, { "concept_id": "r012:concept:numerator", "definition": "The upper component of a fraction.", "prerequisite_concept_ids": [ "r012:concept:fraction" ], "source_label": "numerator", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_rationals" ] }, { "concept_id": "r012:concept:octave", "definition": "A musical interval spanning eight scale degrees and conventionally corresponding to a doubling of frequency.", "prerequisite_concept_ids": [], "source_label": "octave", "status": "source_frozen", "unit_ids": [ "r012:unit:s_polya_apps" ] }, { "concept_id": "r012:concept:odd-part-partition", "definition": "An integer partition in which every positive part is odd.", "prerequisite_concept_ids": [ "r012:concept:integer-partition" ], "source_label": "partition into odd parts", "status": "source_frozen", "unit_ids": [ "r012:unit:fig_genfunction_partsof8", "r012:unit:thm_partition", "r012:unit:auto:c1cbbffbb5bf258ea18be45d", "r012:unit:s_genfunction_discussion", "r012:unit:auto:c53a9727c6ac71ed663831ee", "r012:unit:auto:e0f50e92137238f006185b9f" ] }, { "concept_id": "r012:concept:offline-algorithm", "definition": "An algorithm that may inspect the complete input before making its decisions.", "prerequisite_concept_ids": [], "source_label": "offline algorithm", "status": "source_frozen", "unit_ids": [ "r012:unit:s_kitchensink_online" ] }, { "concept_id": "r012:concept:online-algorithm", "definition": "An algorithm that must make irrevocable decisions as input is revealed sequentially.", "prerequisite_concept_ids": [], "source_label": "online algorithm", "status": "source_frozen", "unit_ids": [ "r012:unit:s_kitchensink_online" ] }, { "concept_id": "r012:concept:online-graph-coloring-game", "definition": "Builder reveals vertices and prior adjacencies while Assigner must immediately choose a proper color.", "prerequisite_concept_ids": [ "r012:concept:online-algorithm", "r012:concept:proper-coloring" ], "source_label": "online graph-coloring game", "status": "source_frozen", "unit_ids": [ "r012:unit:s_kitchensink_online" ] }, { "concept_id": "r012:concept:online-poset-antichain-partition", "definition": "Assign each arriving poset point to an antichain indexed by its longest upward and downward chain lengths.", "prerequisite_concept_ids": [ "r012:concept:online-algorithm", "r012:concept:poset-antichain", "r012:concept:poset-height" ], "source_label": "online poset antichain partition", "status": "source_frozen", "unit_ids": [ "r012:unit:s_kitchensink_online" ] }, { "concept_id": "r012:concept:open-statement", "definition": "A statement involving a variable whose truth depends on the value assigned to that variable.", "prerequisite_concept_ids": [ "r012:concept:positive-integer", "r012:concept:predicate" ], "source_label": "open statement", "status": "source_frozen", "unit_ids": [ "r012:unit:s_induction_induction", "r012:unit:s_background_notation" ] }, { "concept_id": "r012:concept:operating-system", "definition": "System software managing computing hardware and application execution.", "prerequisite_concept_ids": [], "source_label": "operating system", "status": "source_frozen", "unit_ids": [ "r012:unit:ch_epilogue" ] }, { "concept_id": "r012:concept:optimization", "definition": "Selection of a feasible object that maximizes or minimizes an objective.", "prerequisite_concept_ids": [ "r012:concept:algorithm" ], "source_label": "optimization", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_intro", "r012:unit:ch_networkflow", "r012:unit:s_networkflow_intro", "r012:unit:s_networkflow_flows-cuts", "r012:unit:s_networkflow_integer-programming" ] }, { "concept_id": "r012:concept:ordered-coin-sequence-generating-function", "definition": "An ordinary generating function whose coefficients count denomination sequences in which the order of inserted coins matters.", "prerequisite_concept_ids": [ "r012:concept:coin-change-generating-function", "r012:concept:string" ], "source_label": "generating function for ordered coin sequences", "status": "source_frozen", "unit_ids": [ "r012:unit:auto:3a3cdb7defb6ad5b83822a4f", "r012:unit:exer_genfunction_coin-toll", "r012:unit:auto:46be14a19d3b560435dab10c" ] }, { "concept_id": "r012:concept:ordered-pair", "definition": "A two-coordinate object whose coordinate order is significant.", "prerequisite_concept_ids": [], "source_label": "ordered pair", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_products" ] }, { "concept_id": "r012:concept:ordered-tree", "definition": "A rooted tree whose children at every vertex have a specified order.", "prerequisite_concept_ids": [ "r012:concept:rooted-tree" ], "source_label": "ordered tree", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_rubots" ] }, { "concept_id": "r012:concept:ordinary-generating-function", "definition": "A generating function of the form sum from n=0 to infinity of a_n x^n, so the coefficient of x^n is a_n.", "prerequisite_concept_ids": [ "r012:concept:generating-function", "r012:concept:formal-power-series" ], "source_label": "ordinary generating function", "status": "source_frozen", "unit_ids": [ "r012:unit:s_genfunction_exponential", "r012:unit:s_genfunction_exercises", "r012:unit:auto:e3dfb8524ffbb34b5062db90", "r012:unit:auto:a65033c7e640b8b32155ee76", "r012:unit:s_genfunction_PI", "r012:unit:genfunction_PI_exponential-vs-ordinary" ] }, { "concept_id": "r012:concept:oriented-graph", "definition": "A digraph obtained by assigning one direction to every edge of an underlying graph.", "prerequisite_concept_ids": [ "r012:concept:digraph", "r012:concept:graph" ], "source_label": "oriented graph", "status": "source_frozen", "unit_ids": [ "r012:unit:s_graphalgorithms_shortest-paths", "r012:unit:s_networkflow_intro" ] }, { "concept_id": "r012:concept:outcome", "definition": "An elementary member of the outcome set.", "prerequisite_concept_ids": [ "r012:concept:probability-space" ], "source_label": "outcome", "status": "source_frozen", "unit_ids": [ "r012:unit:s_probability_intro" ] }, { "concept_id": "r012:concept:p-class", "definition": "The class of decision problems solvable in polynomial time.", "prerequisite_concept_ids": [ "r012:concept:polynomial-time" ], "source_label": "P", "status": "source_frozen", "unit_ids": [ "r012:unit:s_basics_exact" ] }, { "concept_id": "r012:concept:pair-group", "definition": "The permutation action induced on two-element subsets by permutations of the underlying vertices.", "prerequisite_concept_ids": [ "r012:concept:symmetric-group", "r012:concept:group-action" ], "source_label": "pair group", "status": "source_frozen", "unit_ids": [ "r012:unit:s_polya_apps" ] }, { "concept_id": "r012:concept:pairing-axiom", "definition": "For any two sets there exists a set whose elements are exactly those two sets.", "prerequisite_concept_ids": [ "r012:concept:zfc-set-theory" ], "source_label": "axiom of pairing", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_ZFC" ] }, { "concept_id": "r012:concept:pairwise-disjoint", "definition": "A family of sets in which every two distinct members have empty intersection.", "prerequisite_concept_ids": [], "source_label": "pairwise disjoint", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_enum" ] }, { "concept_id": "r012:concept:pairwise-independent-random-variables", "definition": "Every two distinct real-valued random variables in the family have independent threshold events for all real thresholds.", "prerequisite_concept_ids": [ "r012:concept:random-variable", "r012:concept:independent-events" ], "source_label": "pairwise independent random variables", "status": "source_frozen", "unit_ids": [ "r012:unit:auto:4bd593cb3ed0759f3af2567d" ] }, { "concept_id": "r012:concept:partial-fraction-coefficient-extraction", "definition": "Decomposition of a rational generating function into simpler rational terms with recognizable power-series coefficients.", "prerequisite_concept_ids": [ "r012:concept:generating-function", "r012:concept:formal-power-series" ], "source_label": "coefficient extraction by partial fractions", "status": "source_frozen", "unit_ids": [ "r012:unit:s_genfunction_distributions", "r012:unit:exa_genfunction_partial-fracs", "r012:unit:auto:692d3d5206064b390fb35ace" ] }, { "concept_id": "r012:concept:partial-order", "definition": "A reflexive, antisymmetric, and transitive binary relation on a set.", "prerequisite_concept_ids": [ "r012:concept:binary-relation", "r012:concept:reflexive-relation", "r012:concept:antisymmetric-relation", "r012:concept:transitive-relation" ], "source_label": "partial order", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_intro", "r012:unit:s_kitchensink_zero-one-matrices", "r012:unit:s_background_order" ] }, { "concept_id": "r012:concept:particular-plus-homogeneous-solution", "definition": "If f_0 is one solution of p(A)f=g, every other solution differs from f_0 by a member of the homogeneous solution space.", "prerequisite_concept_ids": [ "r012:concept:homogeneous-recurrence-solution-space", "r012:concept:advancement-operator-equation" ], "source_label": "particular solution plus homogeneous solution", "status": "source_frozen", "unit_ids": [ "r012:unit:lem_particular", "r012:unit:auto:3cdf045c69a328229cf28c05", "r012:unit:s_recurrence_solving_nonhomogeneous", "r012:unit:auto:4a09aa7241b0bf6a931d0955", "r012:unit:auto:2b9501b6517e77e868e24689", "r012:unit:ex_recurrence_lines-solved", "r012:unit:auto:025bf0a72a4db690049c74bf" ] }, { "concept_id": "r012:concept:particular-solution", "definition": "One solution matching the forcing term of a nonhomogeneous recurrence, combined with the homogeneous solution to obtain the general solution.", "prerequisite_concept_ids": [ "r012:concept:nonhomogeneous-linear-recurrence" ], "source_label": "particular solution of a nonhomogeneous recurrence", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_genfunction" ] }, { "concept_id": "r012:concept:particular-solution-trial-method", "definition": "For p(A)f=g, choose a trial form modeled on the right-hand side with undetermined constants, apply p(A), and determine the constants by matching the resulting coefficients.", "prerequisite_concept_ids": [ "r012:concept:nonhomogeneous-linear-recurrence", "r012:concept:advancement-operator-polynomial" ], "source_label": "trial method for a particular solution", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_solving_nonhomogeneous", "r012:unit:auto:4a09aa7241b0bf6a931d0955", "r012:unit:auto:2b9501b6517e77e868e24689", "r012:unit:ex_recurrence_lines-solved", "r012:unit:auto:025bf0a72a4db690049c74bf", "r012:unit:rq-recurrence_solving-async-nonlinear" ] }, { "concept_id": "r012:concept:particular-trial-overlap-adjustment", "definition": "When a proposed forcing-shaped trial contains terms annihilated by the homogeneous operator, modify the trial, commonly by multiplying by a sufficient power of n or raising polynomial degree, before matching coefficients.", "prerequisite_concept_ids": [ "r012:concept:particular-solution-trial-method", "r012:concept:homogeneous-linear-recurrence" ], "source_label": "adjusting a particular trial that overlaps the homogeneous solution", "status": "source_frozen", "unit_ids": [ "r012:unit:auto:2b9501b6517e77e868e24689", "r012:unit:ex_recurrence_lines-solved", "r012:unit:auto:025bf0a72a4db690049c74bf", "r012:unit:rq-recurrence_solving-async-nonlinear" ] }, { "concept_id": "r012:concept:partition", "definition": "A decomposition into mutually disjoint parts that exhaust the original object.", "prerequisite_concept_ids": [ "r012:concept:pairwise-disjoint" ], "source_label": "partition", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_intro", "r012:unit:s_intro_enum" ] }, { "concept_id": "r012:concept:partition-cover-relation", "definition": "A cover transfers one unit from an earlier part to a later part under the stated plateau constraints.", "prerequisite_concept_ids": [ "r012:concept:partition-dominance-order" ], "source_label": "cover relation in partition dominance order", "status": "source_frozen", "unit_ids": [ "r012:unit:prop_partitioncover", "r012:unit:s_kitchensink_zero-one-matrices" ] }, { "concept_id": "r012:concept:partition-dominance-order", "definition": "V dominates W when every relevant prefix sum of V is at least the corresponding prefix sum of W.", "prerequisite_concept_ids": [ "r012:concept:integer-partition" ], "source_label": "dominance order on integer partitions", "status": "source_frozen", "unit_ids": [ "r012:unit:s_kitchensink_zero-one-matrices" ] }, { "concept_id": "r012:concept:partition-generating-function", "definition": "The infinite product over positive part sizes whose coefficient of x^n is the partition number p_n.", "prerequisite_concept_ids": [ "r012:concept:integer-partition", "r012:concept:generating-function", "r012:concept:power-series-product" ], "source_label": "generating function for integer partitions", "status": "source_frozen", "unit_ids": [ "r012:unit:s_genfunction_partitions-integer", "r012:unit:thm_partition", "r012:unit:auto:c1cbbffbb5bf258ea18be45d", "r012:unit:auto:e0f50e92137238f006185b9f", "r012:unit:auto:5a5a5f490053509f506e2e65", "r012:unit:auto:fce61924f5d5515db0dda1a7" ] }, { "concept_id": "r012:concept:partition-number", "definition": "The number p_n of integer partitions of n, with p_0=1 by convention.", "prerequisite_concept_ids": [ "r012:concept:integer-partition", "r012:concept:sequence" ], "source_label": "partition number", "status": "source_frozen", "unit_ids": [ "r012:unit:s_genfunction_partitions-integer" ] }, { "concept_id": "r012:concept:partitioned-event-probability", "definition": "An event split into disjoint cases has probability equal to the sum of the case probabilities.", "prerequisite_concept_ids": [ "r012:concept:conditional-probability", "r012:concept:disjoint-events" ], "source_label": "probability over a partition", "status": "source_frozen", "unit_ids": [ "r012:unit:exa_twojars" ] }, { "concept_id": "r012:concept:pascal-triangle", "definition": "A triangular arrangement of binomial coefficients in which each interior entry is the sum of the two entries above it.", "prerequisite_concept_ids": [ "r012:concept:binomial-coefficient", "r012:concept:recursion" ], "source_label": "Pascal's triangle", "status": "source_frozen", "unit_ids": [ "r012:unit:s_induction_bincoeffs", "r012:unit:fig_pascal" ] }, { "concept_id": "r012:concept:path", "definition": "A sequence of vertices in which consecutive vertices are adjacent.", "prerequisite_concept_ids": [ "r012:concept:adjacency" ], "source_label": "path", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_graph", "r012:unit:auto:21546ae1bdf04f7552481516", "r012:unit:s_networkflow_augmenting", "r012:unit:s_networkflow_labeling-algorithm", "r012:unit:s_networkflow_example", "r012:unit:s_networkflow_exercises" ] }, { "concept_id": "r012:concept:pattern-inventory", "definition": "A generating function whose coefficients count inequivalent patterns with specified color multiplicities.", "prerequisite_concept_ids": [ "r012:concept:polya-enumeration-theorem" ], "source_label": "pattern inventory", "status": "source_frozen", "unit_ids": [ "r012:unit:s_polya_polya" ] }, { "concept_id": "r012:concept:payoff-ratio", "definition": "The ratio between a wager's payout and the amount staked.", "prerequisite_concept_ids": [ "r012:concept:fair-game" ], "source_label": "payoff ratio", "status": "source_frozen", "unit_ids": [ "r012:unit:auto:07a5e2769ec7b1889ef21fcf" ] }, { "concept_id": "r012:concept:peano-postulates", "definition": "Axioms specifying zero, an injective successor, the absence of a predecessor of zero, and induction.", "prerequisite_concept_ids": [ "r012:concept:mathematical-induction" ], "source_label": "Peano postulates", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_formal-num-sys" ] }, { "concept_id": "r012:concept:perfect-graph", "definition": "A graph for which every induced subgraph has chromatic number equal to clique number.", "prerequisite_concept_ids": [ "r012:concept:induced-subgraph", "r012:concept:chromatic-number", "r012:concept:clique-number" ], "source_label": "perfect graph", "status": "source_frozen", "unit_ids": [ "r012:unit:thm_intgraphcol", "r012:unit:s_posets_dilworth-intord" ] }, { "concept_id": "r012:concept:permutation", "definition": "An ordering or rearrangement of distinct positions or objects.", "prerequisite_concept_ids": [ "r012:concept:enumeration" ], "source_label": "permutation", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_intro", "r012:unit:s_strings_permutations", "r012:unit:exa_strings_perm", "r012:unit:prop_strings_permutations", "r012:unit:exa_inclusion-exclusion_prop-derange", "r012:unit:inclusion-exclusion_PI_perms-count", "r012:unit:inclusion-exclusion_PI_perm-prop" ] }, { "concept_id": "r012:concept:permutation-group", "definition": "A set of permutations containing the identity and closed under composition and inverses.", "prerequisite_concept_ids": [ "r012:concept:permutation", "r012:concept:function-composition" ], "source_label": "permutation group", "status": "source_frozen", "unit_ids": [ "r012:unit:s_polya_perm-groups" ] }, { "concept_id": "r012:concept:permutation-multiplication", "definition": "Composition of permutations, evaluated from right to left.", "prerequisite_concept_ids": [ "r012:concept:cycle-notation", "r012:concept:function-composition" ], "source_label": "permutation multiplication", "status": "source_frozen", "unit_ids": [ "r012:unit:s_polya_perm-groups" ] }, { "concept_id": "r012:concept:pigeonhole-principle", "definition": "A function from a larger finite set to a smaller finite set maps at least two distinct inputs to the same output.", "prerequisite_concept_ids": [ "r012:concept:function", "r012:concept:injection" ], "source_label": "Pigeon Hole Principle", "status": "source_frozen", "unit_ids": [ "r012:unit:s_basics_pigeonhole", "r012:unit:prop_pigeon" ] }, { "concept_id": "r012:concept:planar-drawing", "definition": "A crossing-free drawing of a planar graph in the plane.", "prerequisite_concept_ids": [ "r012:concept:planar-graph" ], "source_label": "planar drawing", "status": "source_frozen", "unit_ids": [ "r012:unit:fig_planar", "r012:unit:fig_k4-planar" ] }, { "concept_id": "r012:concept:planar-graph", "definition": "A graph admitting a drawing in the plane with no edge crossings except at shared endpoints.", "prerequisite_concept_ids": [ "r012:concept:graph", "r012:concept:edge-crossing" ], "source_label": "planar graph", "status": "source_frozen", "unit_ids": [ "r012:unit:s_graphs_planar", "r012:unit:fig_planar", "r012:unit:fig_k4-planar" ] }, { "concept_id": "r012:concept:planarity", "definition": "The property of a graph admitting a planar drawing.", "prerequisite_concept_ids": [ "r012:concept:planar-graph", "r012:concept:planar-drawing" ], "source_label": "planarity", "status": "source_frozen", "unit_ids": [ "r012:unit:s_graphs_planar", "r012:unit:thm_kuratowski" ] }, { "concept_id": "r012:concept:polya-counting", "definition": "Enumeration of patterns under group actions using cycle structure.", "prerequisite_concept_ids": [ "r012:concept:enumeration", "r012:concept:symmetry-equivalence" ], "source_label": "Pólya counting", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_intro" ] }, { "concept_id": "r012:concept:polya-enumeration-theorem", "definition": "The theorem that substituting a color inventory into a permutation group's cycle index yields the generating function for inequivalent colorings.", "prerequisite_concept_ids": [ "r012:concept:symmetry-aware-enumeration", "r012:concept:permutation-group", "r012:concept:cycle-index", "r012:concept:inventory-substitution" ], "source_label": "Pólya enumeration theorem", "status": "source_frozen", "unit_ids": [ "r012:unit:ch_polya", "r012:unit:s_polya_polya" ] }, { "concept_id": "r012:concept:polya-exercise-synthesis", "definition": "Integrated practice with cycle notation, group actions, cycle indices, and inventory substitution.", "prerequisite_concept_ids": [ "r012:concept:cycle-notation", "r012:concept:burnside-lemma", "r012:concept:polya-enumeration-theorem" ], "source_label": "Pólya-method exercise synthesis", "status": "source_frozen", "unit_ids": [ "r012:unit:s_polya_exercises" ] }, { "concept_id": "r012:concept:polynomial", "definition": "A finite sum of coefficients multiplied by nonnegative integer powers of a variable.", "prerequisite_concept_ids": [], "source_label": "polynomial", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_complex" ] }, { "concept_id": "r012:concept:polynomial-expansion", "definition": "The expression obtained by multiplying out a polynomial product or power and collecting its terms.", "prerequisite_concept_ids": [], "source_label": "expansion", "status": "source_frozen", "unit_ids": [ "r012:unit:s_strings_binom-thm", "r012:unit:s_strings_multinom", "r012:unit:s_strings_lattice-paths-binomial-thm_activity" ] }, { "concept_id": "r012:concept:polynomial-time", "definition": "Running time bounded above by a fixed polynomial in the input size.", "prerequisite_concept_ids": [ "r012:concept:computational-complexity", "r012:concept:big-o" ], "source_label": "polynomial time", "status": "source_frozen", "unit_ids": [ "r012:unit:s_basics_complexity", "r012:unit:s_basics_big-oh", "r012:unit:s_basics_exact" ] }, { "concept_id": "r012:concept:poset", "definition": "A set equipped with a partial order.", "prerequisite_concept_ids": [ "r012:concept:discrete-structures" ], "source_label": "partially ordered set (poset)", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_intro", "r012:unit:ch_posets", "r012:unit:s_posets_intro", "r012:unit:s_posets_additional-concepts", "r012:unit:s_posets_dilworth" ] }, { "concept_id": "r012:concept:poset-antichain", "definition": "A subset of a poset whose distinct elements are pairwise incomparable.", "prerequisite_concept_ids": [ "r012:concept:poset-comparability", "r012:concept:poset" ], "source_label": "antichain", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_intro", "r012:unit:thm_dualdilworth", "r012:unit:s_flowapplications_chain-partition", "r012:unit:s_kitchensink_online" ] }, { "concept_id": "r012:concept:poset-automorphism", "definition": "An isomorphism from a poset to itself.", "prerequisite_concept_ids": [ "r012:concept:poset-isomorphism" ], "source_label": "poset automorphism", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_additional-concepts" ] }, { "concept_id": "r012:concept:poset-chain", "definition": "A subset of a poset whose elements are pairwise comparable.", "prerequisite_concept_ids": [ "r012:concept:poset-comparability" ], "source_label": "chain", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_intro", "r012:unit:thm_dilworth" ] }, { "concept_id": "r012:concept:poset-comparability", "definition": "Two elements of a poset are comparable when one is less than or equal to the other.", "prerequisite_concept_ids": [ "r012:concept:poset", "r012:concept:partial-order" ], "source_label": "comparable elements", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_intro", "r012:unit:s_posets_additional-concepts" ] }, { "concept_id": "r012:concept:poset-component", "definition": "A maximal connected subposet.", "prerequisite_concept_ids": [ "r012:concept:connected-poset", "r012:concept:subposet" ], "source_label": "component of a poset", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_additional-concepts" ] }, { "concept_id": "r012:concept:poset-dimension", "definition": "The least number of linear extensions whose intersection is the given partial order.", "prerequisite_concept_ids": [ "r012:concept:partial-order", "r012:concept:linear-extension" ], "source_label": "poset dimension", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_exercises" ] }, { "concept_id": "r012:concept:poset-embedding", "definition": "An isomorphism from a poset onto a subposet of another poset.", "prerequisite_concept_ids": [ "r012:concept:poset-isomorphism", "r012:concept:subposet" ], "source_label": "poset embedding", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_additional-concepts" ] }, { "concept_id": "r012:concept:poset-height", "definition": "The maximum cardinality of a chain in a finite poset.", "prerequisite_concept_ids": [ "r012:concept:poset-chain" ], "source_label": "height of a poset", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_intro", "r012:unit:thm_dualdilworth", "r012:unit:fig_height5" ] }, { "concept_id": "r012:concept:poset-incomparability", "definition": "Two elements of a poset are incomparable when neither is less than or equal to the other.", "prerequisite_concept_ids": [ "r012:concept:poset-comparability" ], "source_label": "incomparable elements", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_intro", "r012:unit:fig_compincomp" ] }, { "concept_id": "r012:concept:poset-isomorphism", "definition": "A bijection between ground sets preserving and reflecting the partial order.", "prerequisite_concept_ids": [ "r012:concept:poset", "r012:concept:bijection" ], "source_label": "poset isomorphism", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_additional-concepts" ] }, { "concept_id": "r012:concept:poset-network-reconstruction", "definition": "Recover comparabilities and extremal elements from the unit-capacity prime/double-prime network.", "prerequisite_concept_ids": [ "r012:concept:poset-to-flow-network" ], "source_label": "poset reconstruction from its flow network", "status": "source_frozen", "unit_ids": [ "r012:unit:s_flowapplications_exercises" ] }, { "concept_id": "r012:concept:poset-rank", "definition": "A level of a graded poset consisting of elements having the same rank value.", "prerequisite_concept_ids": [ "r012:concept:partial-order" ], "source_label": "rank", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_subset-lattice", "r012:unit:fig_4cube" ] }, { "concept_id": "r012:concept:poset-to-flow-network", "definition": "Duplicate each poset element into a prime and double-prime vertex and add a unit-capacity middle edge for each strict comparison.", "prerequisite_concept_ids": [ "r012:concept:network-flow", "r012:concept:poset" ], "source_label": "poset to flow-network construction", "status": "source_frozen", "unit_ids": [ "r012:unit:s_flowapplications_chain-partition" ] }, { "concept_id": "r012:concept:poset-width", "definition": "The maximum cardinality of an antichain in a finite poset.", "prerequisite_concept_ids": [ "r012:concept:poset-antichain" ], "source_label": "width of a poset", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_intro", "r012:unit:thm_dilworth", "r012:unit:fig_width7", "r012:unit:ch_flowapplications", "r012:unit:s_flowapplications_chain-partition" ] }, { "concept_id": "r012:concept:positional-base-notation", "definition": "A representation of nonnegative integers as finite strings of digits selected below a chosen base.", "prerequisite_concept_ids": [ "r012:concept:nonnegative-integer", "r012:concept:division-algorithm" ], "source_label": "positional notation in base b", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_decimal" ] }, { "concept_id": "r012:concept:positive-cut", "definition": "A Dedekind cut representing a positive real number.", "prerequisite_concept_ids": [ "r012:concept:dedekind-cut" ], "source_label": "positive Dedekind cut", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_reals" ] }, { "concept_id": "r012:concept:positive-half-binomial-coefficient", "definition": "The closed form for binom(1/2,k) used to extract Catalan coefficients from sqrt(1-4x).", "prerequisite_concept_ids": [ "r012:concept:generalized-binomial-coefficient" ], "source_label": "one-half generalized binomial coefficient", "status": "source_frozen", "unit_ids": [ "r012:unit:l_half-binomial", "r012:unit:thm_catalan-genfunction" ] }, { "concept_id": "r012:concept:positive-integer", "definition": "An integer greater than zero.", "prerequisite_concept_ids": [], "source_label": "positive integer", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_number", "r012:unit:s_inclusion-exclusion_euler-phi", "r012:unit:thm_eulerphi", "r012:unit:prop_num-divisible", "r012:unit:s_background_int-union", "r012:unit:s_background_finite-sets" ] }, { "concept_id": "r012:concept:positive-integer-induction", "definition": "A subset of positive integers is all positive integers when it contains one and is successor-closed.", "prerequisite_concept_ids": [ "r012:concept:mathematical-induction", "r012:concept:positive-integer" ], "source_label": "induction on positive integers", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_decimal" ] }, { "concept_id": "r012:concept:postdoctoral-position", "definition": "A temporary research appointment held after the doctorate.", "prerequisite_concept_ids": [], "source_label": "postdoctoral position", "status": "source_frozen", "unit_ids": [ "r012:unit:ch_epilogue" ] }, { "concept_id": "r012:concept:power-series", "definition": "An expression of the form sum from n=0 to infinity of a_n x^n, determined by its coefficient sequence.", "prerequisite_concept_ids": [ "r012:concept:sequence" ], "source_label": "power series", "status": "source_frozen", "unit_ids": [ "r012:unit:auto:5d06d4ea1ab81098c21c9be0", "r012:unit:s_genfunction_intro", "r012:unit:exam_genfunction_geom-series", "r012:unit:auto:0be40ee4fe111164419b75cb", "r012:unit:prop_genfunction-product", "r012:unit:s_genfunction_distributions", "r012:unit:s_genfunction_gen-bin-thm", "r012:unit:thm_newton-binomial", "r012:unit:s_genfunction_exponential", "r012:unit:auto:a92a089aa7a8dd70c223b7f8" ] }, { "concept_id": "r012:concept:power-series-product", "definition": "The Cauchy product in which the coefficient of x^n is the sum from k=0 to n of a_k b_{n-k}.", "prerequisite_concept_ids": [ "r012:concept:formal-power-series", "r012:concept:generating-function" ], "source_label": "product of power series", "status": "source_frozen", "unit_ids": [ "r012:unit:s_genfunction_intro", "r012:unit:prop_genfunction-product", "r012:unit:s_genfunction_distributions", "r012:unit:s_genfunction_binom-app", "r012:unit:auto:77bb8e40ef7c0def33263edd", "r012:unit:s_genfunction_partitions-integer", "r012:unit:thm_partition", "r012:unit:auto:c1cbbffbb5bf258ea18be45d", "r012:unit:auto:a65033c7e640b8b32155ee76" ] }, { "concept_id": "r012:concept:power-set-axiom", "definition": "Every set has a set containing exactly all of its subsets.", "prerequisite_concept_ids": [ "r012:concept:zfc-set-theory" ], "source_label": "power-set axiom", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_ZFC" ] }, { "concept_id": "r012:concept:predicate", "definition": "An open statement whose truth value depends on one or more variables.", "prerequisite_concept_ids": [], "source_label": "predicate", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_intro", "r012:unit:s_background_notation" ] }, { "concept_id": "r012:concept:preference", "definition": "A comparison in which one alternative is favored over another.", "prerequisite_concept_ids": [], "source_label": "preference", "status": "source_frozen", "unit_ids": [ "r012:unit:s_probability_discussion" ] }, { "concept_id": "r012:concept:prim-algorithm", "definition": "Grow one tree from a root by repeatedly adding a minimum-weight edge leaving the current vertex set.", "prerequisite_concept_ids": [ "r012:concept:safe-minimum-cut-edge" ], "source_label": "Prim's algorithm", "status": "source_frozen", "unit_ids": [ "r012:unit:alg_prim" ] }, { "concept_id": "r012:concept:prime-counting-function", "definition": "The function pi(n) counting primes not exceeding n.", "prerequisite_concept_ids": [ "r012:concept:number-theory", "r012:concept:function" ], "source_label": "prime-counting function", "status": "source_frozen", "unit_ids": [ "r012:unit:s_basics_exact" ] }, { "concept_id": "r012:concept:prime-factorization", "definition": "Expression of a positive integer as a product of prime powers.", "prerequisite_concept_ids": [ "r012:concept:number-theory" ], "source_label": "prime factorization", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_number", "r012:unit:s_inclusion-exclusion_euler-phi", "r012:unit:thm_eulerphi", "r012:unit:auto:5cf98fb376a04e75419df1f9", "r012:unit:auto:78cd58d5f8395977c731d2e1", "r012:unit:s_inclusion-exclusion_discussion" ] }, { "concept_id": "r012:concept:prime-membership-witness", "definition": "A factorization witnesses compositeness, while primality requires a complete justification.", "prerequisite_concept_ids": [ "r012:concept:prime-number", "r012:concept:set-membership" ], "source_label": "witnessing prime membership or nonmembership", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_intro" ] }, { "concept_id": "r012:concept:prime-number", "definition": "A positive integer greater than one whose only positive divisors are one and itself.", "prerequisite_concept_ids": [ "r012:concept:positive-integer" ], "source_label": "prime number", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_intro" ] }, { "concept_id": "r012:concept:probabilistic-method", "definition": "An existence method that proves a desired object exists by showing a random object has the desired property with positive probability.", "prerequisite_concept_ids": [ "r012:concept:probability-space", "r012:concept:expected-value" ], "source_label": "probabilistic method", "status": "source_frozen", "unit_ids": [ "r012:unit:s_ramsey_probability", "r012:unit:s_ramsey_girth", "r012:unit:s_kitchensink_applying-local-lemma" ] }, { "concept_id": "r012:concept:probability", "definition": "A numerical measure of uncertainty assigned to events.", "prerequisite_concept_ids": [], "source_label": "probability", "status": "source_frozen", "unit_ids": [ "r012:unit:ch_probability" ] }, { "concept_id": "r012:concept:probability-exercise-synthesis", "definition": "Exercises combining counting, complements, conditional probability, expectation, and independence.", "prerequisite_concept_ids": [ "r012:concept:probability", "r012:concept:conditional-probability", "r012:concept:expected-value" ], "source_label": "probability exercise synthesis", "status": "source_frozen", "unit_ids": [ "r012:unit:s_probability_exercises" ] }, { "concept_id": "r012:concept:probability-measure", "definition": "A normalized finitely additive function assigning values in [0,1] to events.", "prerequisite_concept_ids": [ "r012:concept:probability-space" ], "source_label": "probability measure", "status": "source_frozen", "unit_ids": [ "r012:unit:s_probability_intro" ] }, { "concept_id": "r012:concept:probability-space", "definition": "A finite outcome set together with a probability measure on all of its subsets.", "prerequisite_concept_ids": [ "r012:concept:finite-set", "r012:concept:set-function" ], "source_label": "probability space", "status": "source_frozen", "unit_ids": [ "r012:unit:s_probability_intro", "r012:unit:s_ramsey_probability", "r012:unit:s_kitchensink_lovasz-local" ] }, { "concept_id": "r012:concept:proper-coloring", "definition": "A vertex coloring in which adjacent vertices receive different colors.", "prerequisite_concept_ids": [ "r012:concept:graph-coloring", "r012:concept:adjacency" ], "source_label": "proper coloring", "status": "source_frozen", "unit_ids": [ "r012:unit:s_graphs_color", "r012:unit:disc-graphs_color-intro", "r012:unit:graphs_PI_coloring-defn", "r012:unit:s_kitchensink_online" ] }, { "concept_id": "r012:concept:proper-subset", "definition": "A subset that omits at least one element of the containing set.", "prerequisite_concept_ids": [ "r012:concept:set-inclusion" ], "source_label": "proper subset", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_int-union" ] }, { "concept_id": "r012:concept:proposition", "definition": "A declarative statement having a truth value.", "prerequisite_concept_ids": [], "source_label": "proposition", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_notation", "r012:unit:s_background_equivalence" ] }, { "concept_id": "r012:concept:prufer-code", "definition": "A length-(n-2) sequence encoding a labeled tree on n vertices by repeatedly deleting the least-labeled leaf and recording its neighbor.", "prerequisite_concept_ids": [ "r012:concept:labeled-tree", "r012:concept:sequence", "r012:concept:leaf" ], "source_label": "Prüfer code", "status": "source_frozen", "unit_ids": [ "r012:unit:thm_cayley", "r012:unit:ex_prufer-code", "r012:unit:ex_prufer-code-reverse" ] }, { "concept_id": "r012:concept:public-key-cryptography", "definition": "Cryptographic methods that use a publicly distributable key together with separate secret information; the discussion highlights the factorization of products of large primes as one important setting.", "prerequisite_concept_ids": [], "source_label": "public key cryptography", "status": "source_frozen", "unit_ids": [ "r012:unit:s_inclusion-exclusion_discussion" ] }, { "concept_id": "r012:concept:quantitative-finance-career", "definition": "A finance career centered on intensive analytical problem solving.", "prerequisite_concept_ids": [], "source_label": "quantitative finance career", "status": "source_frozen", "unit_ids": [ "r012:unit:ch_epilogue" ] }, { "concept_id": "r012:concept:quick-sort", "definition": "A comparison-sorting algorithm that partitions around a pivot and recursively sorts the resulting parts.", "prerequisite_concept_ids": [ "r012:concept:sorting", "r012:concept:recursion" ], "source_label": "quick sort", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_sorting" ] }, { "concept_id": "r012:concept:radius-of-convergence", "definition": "The nonnegative radius R within which a power series converges and beyond which it diverges; R may be zero or infinite.", "prerequisite_concept_ids": [ "r012:concept:power-series" ], "source_label": "radius of convergence", "status": "source_frozen", "unit_ids": [ "r012:unit:s_genfunction_intro" ] }, { "concept_id": "r012:concept:ramsey-event-dependency-count", "definition": "Bad events depend exactly when their underlying vertex sets share at least two vertices, yielding explicit neighborhood counts.", "prerequisite_concept_ids": [ "r012:concept:dependency-neighborhood", "r012:concept:triangle-bad-event", "r012:concept:independent-set-bad-event" ], "source_label": "Ramsey event dependency count", "status": "source_frozen", "unit_ids": [ "r012:unit:s_kitchensink_applying-local-lemma" ] }, { "concept_id": "r012:concept:ramsey-number", "definition": "The least graph order forcing either a clique of one prescribed size or an independent set of another prescribed size.", "prerequisite_concept_ids": [ "r012:concept:clique", "r012:concept:independent-set" ], "source_label": "Ramsey number", "status": "source_frozen", "unit_ids": [ "r012:unit:s_probmeth_graph-ramsey", "r012:unit:s_ramsey_small", "r012:unit:s_kitchensink_applying-local-lemma" ] }, { "concept_id": "r012:concept:ramsey-property", "definition": "A threshold property guaranteeing a clique or an independent set of a prescribed size.", "prerequisite_concept_ids": [ "r012:concept:clique", "r012:concept:independent-set" ], "source_label": "clique-or-independent-set guarantee", "status": "source_frozen", "unit_ids": [ "r012:unit:auto:4f427a0d8cd35dd9d5f9a181", "r012:unit:thm_graphramsey", "r012:unit:thm_genramsey", "r012:unit:s_kitchensink_arithmetic-combinatorics", "r012:unit:auto:2c43873e0968cfc6d26f9acb" ] }, { "concept_id": "r012:concept:random-graph", "definition": "A graph generated by a specified probability model.", "prerequisite_concept_ids": [ "r012:concept:graph", "r012:concept:probability-space" ], "source_label": "random graph", "status": "source_frozen", "unit_ids": [ "r012:unit:auto:6691548129df37fcc64c5423", "r012:unit:s_ramsey_probability", "r012:unit:s_ramsey_girth", "r012:unit:s_ramsey_exercises", "r012:unit:s_kitchensink_applying-local-lemma" ] }, { "concept_id": "r012:concept:random-tournament", "definition": "A tournament obtained by independently choosing each edge orientation with equal probability.", "prerequisite_concept_ids": [ "r012:concept:tournament", "r012:concept:probability-space" ], "source_label": "random tournament", "status": "source_frozen", "unit_ids": [ "r012:unit:s_ramsey_exercises" ] }, { "concept_id": "r012:concept:random-variable", "definition": "A real-valued function on the outcomes of a probability space.", "prerequisite_concept_ids": [ "r012:concept:probability-space", "r012:concept:function" ], "source_label": "random variable", "status": "source_frozen", "unit_ids": [ "r012:unit:auto:fa26c80a67c89c43da03624e", "r012:unit:s_ramsey_probability", "r012:unit:thm_girth" ] }, { "concept_id": "r012:concept:rational-arithmetic", "definition": "Well-defined arithmetic on fraction representatives using common-denominator addition and componentwise multiplication.", "prerequisite_concept_ids": [ "r012:concept:rational-number", "r012:concept:well-defined-operation" ], "source_label": "rational addition and multiplication", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_rationals" ] }, { "concept_id": "r012:concept:rational-cut", "definition": "The cut of all rational numbers strictly below a specified rational q.", "prerequisite_concept_ids": [ "r012:concept:dedekind-cut", "r012:concept:rational-number" ], "source_label": "rational cut", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_reals" ] }, { "concept_id": "r012:concept:rational-number", "definition": "An equivalence class represented by an integer numerator and nonzero integer denominator.", "prerequisite_concept_ids": [ "r012:concept:rational-pair-equivalence", "r012:concept:equivalence-class" ], "source_label": "rational number", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_rationals" ] }, { "concept_id": "r012:concept:rational-order", "definition": "For positive denominators, (a,b) is at most (c,d) when ad is at most bc.", "prerequisite_concept_ids": [ "r012:concept:rational-number", "r012:concept:integer-total-order" ], "source_label": "total order on the rationals", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_rationals" ] }, { "concept_id": "r012:concept:rational-pair-equivalence", "definition": "Integer pairs (x,y) and (z,w), with nonzero denominators, are equivalent when xw=yz.", "prerequisite_concept_ids": [ "r012:concept:integer", "r012:concept:equivalence-relation" ], "source_label": "fraction-pair equivalence", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_rationals" ] }, { "concept_id": "r012:concept:real-number-as-cut", "definition": "A real number represented by a Dedekind cut of the rationals.", "prerequisite_concept_ids": [ "r012:concept:dedekind-cut" ], "source_label": "real number as a cut", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_reals" ] }, { "concept_id": "r012:concept:real-vector-space", "definition": "A set of vectors equipped with addition and real scalar multiplication satisfying the eight listed vector-space axioms.", "prerequisite_concept_ids": [], "source_label": "vector space over the real numbers", "status": "source_frozen", "unit_ids": [ "r012:unit:auto:ba2f054b693f6319b15736f2" ] }, { "concept_id": "r012:concept:receiver-monotonicity", "definition": "The receiver retained by each seller weakly improves throughout deferred acceptance.", "prerequisite_concept_ids": [ "r012:concept:deferred-acceptance-algorithm" ], "source_label": "receiver-side monotonicity", "status": "source_frozen", "unit_ids": [ "r012:unit:s_kitchensink_stable-match" ] }, { "concept_id": "r012:concept:recurrence", "definition": "A rule that specifies later terms or function values from earlier ones, together with enough initial values to determine the sequence.", "prerequisite_concept_ids": [ "r012:concept:sequence" ], "source_label": "recurrence", "status": "source_frozen", "unit_ids": [ "r012:unit:ch_recurrence", "r012:unit:auto:b2a4979434434bc7233ca327", "r012:unit:s_recurrence_intro", "r012:unit:s_recurrence_intro_fib", "r012:unit:auto:a5fd615b1106d334b47d57d4", "r012:unit:s_recurrence_intro_strings", "r012:unit:ex_recurrence_binary-strings-no-11", "r012:unit:ex_recurrence_ternary-strings-no-20", "r012:unit:s_recurrence_intro_lines", "r012:unit:s_recurrence_solving", "r012:unit:s_recurrence_solving_homogeneous", "r012:unit:s_recurrence_solving_nonhomogeneous" ] }, { "concept_id": "r012:concept:recurrence-equation", "definition": "An equation expressing a sequence term or function value in terms of preceding terms or values.", "prerequisite_concept_ids": [ "r012:concept:recurrence" ], "source_label": "recurrence equation", "status": "source_frozen", "unit_ids": [ "r012:unit:ch_recurrence", "r012:unit:auto:b2a4979434434bc7233ca327", "r012:unit:s_recurrence_intro", "r012:unit:s_recurrence_solving", "r012:unit:ex_recurrence_deg2", "r012:unit:ex_recurrence_ternary-strings-no-20-solved", "r012:unit:ex_recurrence_deg4-repeated", "r012:unit:s_recurrence_solving_nonhomogeneous", "r012:unit:ex_recurrence_lines-solved", "r012:unit:rqs-recurrence_solving" ] }, { "concept_id": "r012:concept:recurrence-equation-exercise-synthesis", "definition": "Integrated practice deriving, classifying, and solving recurrence equations with varied initial data.", "prerequisite_concept_ids": [ "r012:concept:recurrence-relation", "r012:concept:linear-recurrence-equation" ], "source_label": "recurrence equation exercise synthesis", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_exercises" ] }, { "concept_id": "r012:concept:recurrence-generating-function-exercise-synthesis", "definition": "Practice solving homogeneous and nonhomogeneous recurrences through generating functions and extracting closed forms.", "prerequisite_concept_ids": [ "r012:concept:generating-function-recurrence-method" ], "source_label": "generating-function exercise synthesis for recurrences", "status": "source_frozen", "unit_ids": [ "r012:unit:ex_recurrence_gen-fn", "r012:unit:auto:fdcb4e382775725188f7ddfb", "r012:unit:auto:8b87a353e8228ac6d2ab68b5", "r012:unit:auto:3110fa2523dd10b3fb16b771" ] }, { "concept_id": "r012:concept:recurrence-relation", "definition": "A relation expressing sequence terms through earlier terms.", "prerequisite_concept_ids": [ "r012:concept:enumeration" ], "source_label": "recurrence relation", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_intro", "r012:unit:s_recurrence_adv-ops", "r012:unit:ex_recurrence_adveq", "r012:unit:s_recurrence_adv-ops_const-coeff", "r012:unit:rqs-recurrence_adv-ops", "r012:unit:rq-recurrence_adv-ops_rewrite" ] }, { "concept_id": "r012:concept:recurrence-runtime-benchmark", "definition": "An empirical timing comparison between direct recursion and a mathematically equivalent explicit formula.", "prerequisite_concept_ids": [ "r012:concept:recursive-evaluation", "r012:concept:explicit-recurrence-solution" ], "source_label": "runtime comparison of recursive and explicit recurrence evaluation", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_discussion", "r012:unit:recurrence_discussion_takeaways" ] }, { "concept_id": "r012:concept:recursion", "definition": "The specification or computation of an object in terms of smaller instances of the same kind.", "prerequisite_concept_ids": [ "r012:concept:sequence" ], "source_label": "recursion", "status": "source_frozen", "unit_ids": [ "r012:unit:ch_induction" ] }, { "concept_id": "r012:concept:recursive-addition", "definition": "Addition on natural numbers defined by n+0=n and n+s(k)=s(n+k).", "prerequisite_concept_ids": [ "r012:concept:successor-function", "r012:concept:mathematical-induction" ], "source_label": "formal recursive definition of addition", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_add", "r012:unit:s_background_formal-num-sys" ] }, { "concept_id": "r012:concept:recursive-evaluation", "definition": "Direct computation of a sequence value by recursively invoking earlier sequence values.", "prerequisite_concept_ids": [ "r012:concept:recursion", 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"r012:concept:binary-relation" ], "source_label": "reflexive binary relation", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_order", "r012:unit:s_background_equivalence", "r012:unit:s_background_integers" ] }, { "concept_id": "r012:concept:regular-graph", "definition": "A graph in which every vertex has the same degree.", "prerequisite_concept_ids": [ "r012:concept:graph" ], "source_label": "regular graph", "status": "source_frozen", "unit_ids": [ "r012:unit:ch_polya" ] }, { "concept_id": "r012:concept:regular-transition-matrix", "definition": "A transition matrix for which some positive power has only positive entries.", "prerequisite_concept_ids": [ "r012:concept:transition-matrix" ], "source_label": "regular transition matrix", "status": "source_frozen", "unit_ids": [ "r012:unit:thm_regular", "r012:unit:s_kitchensink_markov-chains" ] }, { "concept_id": "r012:concept:regularity-axiom", "definition": "Every nonempty set has an element disjoint from it.", "prerequisite_concept_ids": [ "r012:concept:zfc-set-theory" ], "source_label": "axiom of regularity", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_ZFC" ] }, { "concept_id": "r012:concept:relative-primality", "definition": "Two integers are relatively prime when their greatest common divisor is 1.", "prerequisite_concept_ids": [ "r012:concept:greatest-common-divisor" ], "source_label": "relatively prime", "status": "source_frozen", "unit_ids": [ "r012:unit:s_inclusion-exclusion_euler-phi", "r012:unit:inclusion-exclusion_PI_rel-prime" ] }, { "concept_id": "r012:concept:repeated-independent-sampling", "definition": "Repeated draws from one probability space in which each draw is independent of the preceding draws.", "prerequisite_concept_ids": [ "r012:concept:independent-events", "r012:concept:probability-space" ], "source_label": "repeated independent sampling", "status": "source_frozen", "unit_ids": [ "r012:unit:auto:9f5b75658465146534f073d6" ] }, { "concept_id": "r012:concept:repeated-root-recurrence-solution", "definition": "A root r of multiplicity k contributes the k independent solution forms r^n, nr^n, through n^(k-1)r^n.", "prerequisite_concept_ids": [ "r012:concept:distinct-root-recurrence-solution" ], "source_label": "repeated-root solution of a homogeneous recurrence", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_rigorous_repeated", "r012:unit:lem_rr", "r012:unit:eqn_rr", "r012:unit:eqn_solutionrr", "r012:unit:ex_recurrence_deg2-repeated", "r012:unit:ex_recurrence_deg4-repeated" ] }, { "concept_id": "r012:concept:repeating-decimal", "definition": "A decimal expansion with a finite block of digits that repeats indefinitely.", "prerequisite_concept_ids": [ "r012:concept:decimal-expansion", "r012:concept:rational-number" ], "source_label": "repeating decimal", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_complex" ] }, { "concept_id": "r012:concept:replacement-axiom", "definition": "The image of a set under a definable functional rule is a set.", "prerequisite_concept_ids": [ "r012:concept:zfc-set-theory" ], "source_label": "axiom schema of replacement", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_ZFC" ] }, { "concept_id": "r012:concept:required-symbol-egf-factor", "definition": "The factor e^x-1, or a corresponding truncated exponential series, used to require one or more occurrences of a symbol.", "prerequisite_concept_ids": [ "r012:concept:exponential-generating-function" ], "source_label": "required-symbol exponential generating-function factor", "status": "source_frozen", "unit_ids": [ "r012:unit:auto:3259ae9833c0290eb6d05545", "r012:unit:auto:4b9b577c7e32e0494cb91c36", "r012:unit:ex_genfunction_exponential_passcode", "r012:unit:rqs-genfunction_exponential", "r012:unit:rq-genfunction_exponential_passcode", "r012:unit:auto:cd91522cddb9cb10442a7836", "r012:unit:auto:f3edcd631926927e1e6b9719", "r012:unit:auto:fdf77306cd6d226cfda10c55", "r012:unit:auto:39ab9828f6c93359abfbc514", "r012:unit:genfunction_PI_string-restrict-4", "r012:unit:genfunction_PI_string-restrict-2" ] }, { "concept_id": "r012:concept:restricted-exponentiation", "definition": "A two-argument natural-number operation defined on all pairs except (0,0), recursively in the exponent.", "prerequisite_concept_ids": [ "r012:concept:recursive-multiplication" ], "source_label": "exponentiation excluding 0^0", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_exp" ] }, { "concept_id": "r012:concept:restricted-object-distribution", "definition": "A distribution problem whose recipient or object-type counts obey lower bounds, upper bounds, parity, congruence, or other allowed-value restrictions.", "prerequisite_concept_ids": [ "r012:concept:generating-function-distribution-model" ], "source_label": "restricted distribution of objects", "status": "source_frozen", "unit_ids": [ "r012:unit:s_genfunction_distributions", "r012:unit:exa_genfunction_apples", "r012:unit:exa_genfunction_fruit", "r012:unit:auto:2d043fada756b0e678fbcfa4", "r012:unit:exa_genfunction_partial-fracs", "r012:unit:auto:692d3d5206064b390fb35ace", "r012:unit:rqs-genfunction_distributions", "r012:unit:genfunction_distributions_rq-constant_term", "r012:unit:genfunction_distributions_rq-coins", "r012:unit:auto:e1321354951ba5fcec2ae15c", "r012:unit:auto:ad5800950b29d7f51c43a73e", "r012:unit:ex_genfunction_ineq-clean", "r012:unit:auto:6b0d982902d4e6cfc0dfed45", "r012:unit:auto:318098a808461c4b431eaeec", "r012:unit:auto:9da8ebe78f63c221b5b6bc64", "r012:unit:auto:8def6d29b55112ab987c47ed", "r012:unit:ex_genfunction_ineq-messy", "r012:unit:genfunction_PI_apples-one", "r012:unit:genfunction_PI_apples-five", "r012:unit:genfunction_PI_apples-oranges-bananas", "r012:unit:genfunction_PI_polynomial" ] }, { "concept_id": "r012:concept:restricted-passcode-egf", "definition": "A product of exponential-series factors encoding independent parity, lower-bound, and upper-bound restrictions on passcode digits.", "prerequisite_concept_ids": [ "r012:concept:exponential-generating-function", "r012:concept:even-symbol-count-filter", "r012:concept:required-symbol-egf-factor" ], "source_label": "exponential generating function for restricted passcodes", "status": "source_frozen", "unit_ids": [ "r012:unit:ex_genfunction_exponential_passcode", "r012:unit:rqs-genfunction_exponential", "r012:unit:rq-genfunction_exponential_passcode" ] }, { "concept_id": "r012:concept:right-child", "definition": "The second of the two ordered children at an internal RUBOT vertex.", "prerequisite_concept_ids": [ "r012:concept:ordered-tree", "r012:concept:tree-child" ], "source_label": "right child", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_rubots" ] }, { "concept_id": "r012:concept:root-vertex", "definition": "A designated vertex from which a graph construction or path computation begins.", "prerequisite_concept_ids": [ "r012:concept:vertex" ], "source_label": "root vertex", "status": "source_frozen", "unit_ids": [ "r012:unit:ch_graphalgorithms" ] }, { "concept_id": "r012:concept:rooted-tree", "definition": "A tree with one designated vertex called its root.", "prerequisite_concept_ids": [ "r012:concept:tree", "r012:concept:vertex" ], "source_label": "rooted tree", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_rubots" ] }, { "concept_id": "r012:concept:row-column-sum-strings", "definition": "The ordered row totals and column totals of a zero-one matrix.", "prerequisite_concept_ids": [ "r012:concept:zero-one-matrix" ], "source_label": "row and column sum strings", "status": "source_frozen", "unit_ids": [ "r012:unit:s_kitchensink_zero-one-matrices" ] }, { "concept_id": "r012:concept:rubot", "definition": "A rooted, unlabeled, full binary, ordered tree, abbreviated RUBOT.", "prerequisite_concept_ids": [ "r012:concept:rooted-tree", "r012:concept:unlabeled-tree", "r012:concept:full-binary-tree", "r012:concept:ordered-tree" ], "source_label": "rooted unlabeled binary ordered tree", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_rubots", "r012:unit:fig_rubots", "r012:unit:thm_catalan-genfunction" ] }, { "concept_id": "r012:concept:rubot-catalan-recurrence", "definition": "The decomposition of a RUBOT at its ordered left and right subtrees, giving c_n=sum from k=0 to n of c_k c_(n-k) for n at least 2 with c_0=0 and c_1=1.", "prerequisite_concept_ids": [ "r012:concept:rubot", "r012:concept:recurrence-relation" ], "source_label": "Catalan recurrence for RUBOTs", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_rubots" ] }, { "concept_id": "r012:concept:rubot-generating-function", "definition": "The Catalan generating function C(x)=(1-sqrt(1-4x))/2 whose x^n coefficient counts RUBOTs with n leaves.", "prerequisite_concept_ids": [ "r012:concept:rubot-catalan-recurrence", "r012:concept:generating-function", "r012:concept:generalized-binomial-theorem" ], "source_label": "generating function for RUBOTs", "status": "source_frozen", "unit_ids": [ "r012:unit:s_recurrence_rubots", "r012:unit:thm_catalan-genfunction" ] }, { "concept_id": "r012:concept:safe-minimum-cut-edge", "definition": "A minimum-weight edge leaving a current forest component belongs to some minimum-weight spanning tree extending that forest.", "prerequisite_concept_ids": [ "r012:concept:spanning-tree-exchange-principle" ], "source_label": "minimum edge leaving a forest component", "status": "source_frozen", "unit_ids": [ "r012:unit:lem_graphalgorithms_tech" ] }, { "concept_id": "r012:concept:safety-net", "definition": "An institutional protective system supporting public security and well-being.", "prerequisite_concept_ids": [ "r012:concept:computational-public-health" ], "source_label": "safety net", "status": "source_frozen", "unit_ids": [ "r012:unit:ch_epilogue" ] }, { "concept_id": "r012:concept:sage-power-series-expansion", "definition": "Use of SageMath series and coefficient-list operations to expand a generating function and extract selected coefficients.", "prerequisite_concept_ids": [ "r012:concept:power-series", "r012:concept:partial-fraction-coefficient-extraction" ], "source_label": "SageMath power-series expansion", "status": "source_frozen", "unit_ids": [ "r012:unit:s_genfunction_distributions", "r012:unit:auto:692d3d5206064b390fb35ace", "r012:unit:ex_genfunction_exponential_passcode" ] }, { "concept_id": "r012:concept:sample-space", "definition": "The set of all outcomes of an experiment.", "prerequisite_concept_ids": [], "source_label": "sample space", "status": "source_frozen", "unit_ids": [ "r012:unit:s_probability_intro" ] }, { "concept_id": "r012:concept:sampling-without-replacement", "definition": "Random selection in which selected objects are not returned before later selections.", "prerequisite_concept_ids": [ "r012:concept:probability" ], "source_label": "sampling without replacement", "status": "source_frozen", "unit_ids": [ "r012:unit:auto:d275dcdb8b05d03219dfcac9" ] }, { "concept_id": "r012:concept:scale", "definition": "An ordered collection of pitches or pitch classes used as a tonal framework.", "prerequisite_concept_ids": [], "source_label": "musical scale", "status": "source_frozen", "unit_ids": [ "r012:unit:s_polya_apps" ] }, { "concept_id": "r012:concept:scale-transposition", "definition": "Uniform addition modulo twelve to every pitch class in a scale.", "prerequisite_concept_ids": [ "r012:concept:cyclic-group", "r012:concept:modular-arithmetic" ], "source_label": "scale transposition", "status": "source_frozen", "unit_ids": [ "r012:unit:s_polya_apps" ] }, { "concept_id": "r012:concept:scheduling-irregularity", "definition": "A disruption requiring changes to an operating schedule.", "prerequisite_concept_ids": [ "r012:concept:industrial-optimization" ], "source_label": "scheduling irregularity", "status": "source_frozen", "unit_ids": [ "r012:unit:ch_epilogue" ] }, { "concept_id": "r012:concept:second-coordinate", "definition": "The second component of an ordered pair.", "prerequisite_concept_ids": [ "r012:concept:ordered-pair" ], "source_label": "second coordinate", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_products" ] }, { "concept_id": "r012:concept:self-dual-poset", "definition": "A poset isomorphic to its dual.", "prerequisite_concept_ids": [ "r012:concept:dual-poset", "r012:concept:poset-isomorphism" ], "source_label": "self-dual poset", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_additional-concepts" ] }, { "concept_id": "r012:concept:separation-axiom", "definition": "A definable subclass of a set is itself a set.", "prerequisite_concept_ids": [ "r012:concept:zfc-set-theory" ], "source_label": "axiom schema of separation", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_ZFC" ] }, { "concept_id": "r012:concept:sequence", "definition": "An ordered list of mathematical objects indexed by position.", "prerequisite_concept_ids": [], "source_label": "sequence", "status": "source_frozen", "unit_ids": [ "r012:unit:ex_collatz", "r012:unit:thm_ErdosSzekeres", "r012:unit:s_recurrence_intro_fib", "r012:unit:tab_fib-ratios", "r012:unit:auto:12b6629772d99bdcaab15a9b", "r012:unit:ex_recurrence_adveq" ] }, { "concept_id": "r012:concept:set", "definition": "A collection whose membership is unambiguous for every candidate element.", "prerequisite_concept_ids": [], "source_label": "set", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_intro" ] }, { "concept_id": "r012:concept:set-builder-notation", "definition": "Notation defining a set through a membership rule rather than an explicit list.", "prerequisite_concept_ids": [ "r012:concept:set", "r012:concept:predicate" ], "source_label": "set-builder notation", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_intro" ] }, { "concept_id": "r012:concept:set-equality-criterion", "definition": "Two sets are equal exactly when each is a subset of the other.", "prerequisite_concept_ids": [ "r012:concept:set-inclusion" ], "source_label": "equality of sets", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_int-union" ] }, { "concept_id": "r012:concept:set-family", "definition": "A collection of sets considered as one mathematical object.", "prerequisite_concept_ids": [], "source_label": "set family", "status": "source_frozen", "unit_ids": [ "r012:unit:s_kitchensink_extremal-set-theory" ] }, { "concept_id": "r012:concept:set-function", "definition": "A function whose arguments are sets.", "prerequisite_concept_ids": [ "r012:concept:function" ], "source_label": "set function", "status": "source_frozen", "unit_ids": [ "r012:unit:s_probability_intro" ] }, { "concept_id": "r012:concept:set-inclusion", "definition": "The partial order in which one set precedes another exactly when it is a subset of the other.", "prerequisite_concept_ids": [ "r012:concept:set-membership", "r012:concept:partial-order" ], "source_label": "set inclusion", "status": "source_frozen", "unit_ids": [ "r012:unit:s_posets_intro", "r012:unit:s_posets_subset-lattice", "r012:unit:s_background_int-union", "r012:unit:s_background_relations" ] }, { "concept_id": "r012:concept:set-intersection", "definition": "The set of elements belonging to both given sets.", "prerequisite_concept_ids": [ "r012:concept:set-membership" ], "source_label": "set intersection", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_int-union" ] }, { "concept_id": "r012:concept:set-membership", "definition": "The relation in which a specified object is an element of a set.", "prerequisite_concept_ids": [ "r012:concept:set" ], "source_label": "contain", "status": "source_frozen", "unit_ids": [ "r012:unit:induction_PI_subsets-contain-1", "r012:unit:s_background_intro", "r012:unit:s_background_int-union", "r012:unit:s_background_relations", "r012:unit:s_background_finite-sets" ] }, { "concept_id": "r012:concept:set-partition", "definition": "A family of pairwise disjoint subsets whose union is the original set.", "prerequisite_concept_ids": [ "r012:concept:pairwise-disjoint" ], "source_label": "partition of a set", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_enum", "r012:unit:s_background_equivalence" ] }, { "concept_id": "r012:concept:set-theoretic-foundations", "definition": "An axiomatic framework in which mathematical objects and number systems are constructed from sets.", "prerequisite_concept_ids": [], "source_label": "set-theoretic foundations", "status": "source_frozen", "unit_ids": [ "r012:unit:app_background", "r012:unit:s_background_ZFC" ] }, { "concept_id": "r012:concept:set-union", "definition": "The set of elements belonging to at least one of the given sets.", "prerequisite_concept_ids": [ "r012:concept:set-membership" ], "source_label": "set union", "status": "source_frozen", "unit_ids": [ "r012:unit:s_background_int-union" ] }, { "concept_id": "r012:concept:shortest-path", "definition": "A path of minimum total weight between specified vertices.", "prerequisite_concept_ids": [ "r012:concept:path", "r012:concept:weighted-graph", "r012:concept:optimization" ], "source_label": "shortest path", "status": "source_frozen", "unit_ids": [ "r012:unit:s_intro_opt" ] }, { "concept_id": "r012:concept:shortest-path-algorithm-history", "definition": "The publication lineage connecting Moore's equivalent form and Dijkstra's shortest-path algorithm.", "prerequisite_concept_ids": [ "r012:concept:dijkstra-algorithm" ], "source_label": "history of shortest-path algorithms", "status": "source_frozen", "unit_ids": [ "r012:unit:s_graphalgorithms_historical-notes" ] }, { "concept_id": "r012:concept:shortest-path-subpath-optimality", "definition": "Every prefix and suffix cut at an internal vertex of a shortest path is itself shortest between its endpoints.", "prerequisite_concept_ids": [ 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